Unit III: Lec 2 | MIT Calculus Revisited: Single Variable Calculus

The paragraph introduces the concept of inverse trigonometric functions, emphasizing the need for a function to be one-to-one (1:1) to have an inverse. The speaker explains that while the sine function is not naturally 1:1, it can be restricted to certain intervals where it is 1:1, allowing for the definition of inverse functions. The concept of 's sub 0 of x' is introduced as a restricted version of the sine function, and the importance of domain in defining functions is highlighted. The paragraph sets the stage for a deeper exploration of inverse trigonometric functions.

This paragraph delves into the specifics of the inverse sine function, discussing how it can be derived from the sine function by restricting its domain to intervals where it is 1:1. The speaker introduces the idea of 'principal values' and explains how texts often define the inverse sine function with a restricted range for 'y', between -π/2 and π/2. The paragraph also touches on the historical approach to multi-valued functions in older mathematics and how modern definitions require a 1:1 correspondence for inverse functions to exist.

The speaker demonstrates how to find the derivative of the inverse sine function with respect to 'x'. By using the relationship between the sine function and its inverse, and understanding the reciprocal relationship between 'dx/dy' and 'dy/dx' on a 1:1 curve, the derivative is found to be '1 over the square root of 1 - x squared'. The paragraph also discusses the importance of the restricted range for 'y' in ensuring the cosine function's positivity, which simplifies the derivative to a single-valued expression.

The paragraph highlights the utility of inverse trigonometric functions in solving calculus problems that may not initially appear to involve trigonometry. The speaker provides an example of an integral problem whose solution involves the inverse sine function, even though the problem statement does not explicitly mention trigonometric functions. This underscores the importance and broad applicability of inverse trigonometric functions in various mathematical contexts.

This paragraph introduces the technique of trigonometric substitution as a method for solving integrals, particularly those involving the square root of a difference of squares. The speaker illustrates how a right triangle can be used to express the integral in terms of trigonometric functions, leading to a simplification that results in the inverse sine function. The technique is presented as a powerful tool for integrating certain types of functions and is connected to the broader theme of inverse trigonometric functions.

The speaker discusses the process of understanding and deriving identities and derivatives for inverse trigonometric functions. They provide an example of how to visualize and understand the identity involving the inverse cosine and inverse sine functions. The paragraph emphasizes the importance of being familiar with the language of inverse functions and how reducing complex identities to simpler terms can aid in understanding and calculation. The speaker also touches on the derivative of the inverse cosine function, showing how it can be derived from previously established results.

In the concluding paragraph, the speaker summarizes the importance of inverse trigonometric functions and encourages students to explore the material further on their own. They remind students that the key to understanding these functions lies in recognizing the necessity of a 1:1 relationship for defining inverse functions and that the language of 'principal values' is simply a way to ensure this. The speaker also acknowledges the support provided by the Gabriella and Paul Rosenbaum Foundation for the publication of the video and invites viewers to support MIT OpenCourseWare.